Bertrand Maury

• University of Paris-Sud

Understanding the mechanisms behind the remote triggering of landslides by seismic waves at micro-strain amplitude is essential for quantifying seismic hazards. Granular materials provide a relevant model system to investigate landslides within the unjamming transition framework, from solid to liquid states. Furthermore, recent laboratory experimen...

When attracted by a stimulus (light), microswimmers can build up very densely at a constriction and thus cause jamming. The microalga Chlamydomonas Reinhardtii is used here as a model system to study this phenomenon. Its negative phototaxis makes the algae swim away from a light source and go through a microfabricated bottleneck-shaped constriction...

The magnitude and distribution of strain imposed on the peripheral airspaces by mechanical ventilation at the microscopic level and the consequent deformations are unknown despite their importance for understanding the mechanisms occurring at the onset of ventilator-induced lung injury. Here a 4-Dimensional (3D + time) image acquisition and process...

We propose a new microscopic crowd motion model based on Game-Theoretic principles, from which we derive an Inhibition-Based model for evacuation situations. Each individual is supposed to have a desired velocity that they adapt to the behavior of neighbors that influence them. Possible adapted velocities are defined as instantaneous Nash equilibri...

The magnitude and distribution of strain imposed on the peripheral airspaces by mechanical ventilation at the microscopic level and the consequent deformations are unknown despite their importance for understanding the mechanisms occurring at the onset of ventilator-induced lung injury. Here a 4-Dimensional (3D + time) image acquisition and process...

These lecture notes address mathematical issues related to the modeling of impact laws for systems of rigid spheres and their macroscopic counterpart. We analyze the so-called Moreau’s approach to define multibody impact laws at the mircroscopic level, and we analyze the formal macroscopic extensions of these laws, where the non-overlapping constra...

When attracted by a stimulus (e. g. light), microswimmers can build up very densely at a constriction and thus cause clogging. The micro-alga \textit{Chlamydomonas Reinhardtii} is used here as a model system to study this phenomenon. Its negative phototaxis makes the algae swim away from a light source and go through a microfabricated bottleneck-sh...

We propose here a general framework to estimate global evacuation times of complex buildings, and to dynamically investigate the dependence of this evacuation time upon various factors. This model relies on a network, which is in some way the skeleton of the building, the nodes of which are the bottlenecks or exit doors. Those nodes are connected b...

In this paper, we present a numerical method which performs the direct simulation of 2D viscoelastic suspensions. Objectives. Interactions between viscoelastic fluids of Oldroyd type, including inertial effects, and a large number of rigid disks or ellipsoids are addressed. Method. The numerical method is built upon a fictitious domain approach whi...

Cell adhesion on the vascular wall is a highly coupled process where blood flow and adhesion dynamics are closely linked. Cell dynamics in the vicinity of the vascular wall is driven mechanically by the competition between the drag force of the blood flow and the force exerted by the bonds created between the cell and the wall. Bonds exert a fricti...

This paper establishes a link between some space discretization strategies of the Finite Volume type for the Fokker-Planck equation in general meshes (Voronoï tesselations) and gradient flows on the underlying networks of cells, in the framework of discrete Wasserstein distances on graphs recently proposed by Maas [6].

In this chapter, we consider the pressureless Euler equations with a congestion constraint. This system still raises many open questions and, aside from its onedimensional version, very little is known concerning its solutions. The strategy that we propose relies on previous works on crowd motion models with congestion in the framework of the Wasse...

We are interested in this paper in the modelling and numerical simulation of some phenomena that are observed in the context of car dynamics, in particular the appearance of persistent jams upstream critical points with no real cause of flux limitation. We shall consider the case of a stable jam on a freeway upstream an accident that took place on...

We propose and analyze a natural extension of the Moreau sweeping process: given a family of moving convex sets (C(t))t, we look for the evolution of a probability density Pt, constrained to be supported on C(t). We describe in detail three cases: in the first, particles do not interact with each other and stay at rest unless pushed by the moving b...

We propose and analyze a natural extension of the Moreau sweeping process: given a family of moving convex sets $(C(t))_t$, we look for the evolution of a probability density $\rho_t$, constrained to be supported on $C(t)$. We describe in detail three cases: in the first particles do not interact with each other and stay at rest unless pushed by th...

In order to observe growth phenomena in biology where dendritic shapes appear, we propose a simple model where a given population evolves feeded by a diffusing nutriment, but is subject to a density constraint. The particles (e.g., cells) of the population spontaneously stay passive at rest, and only move in order to satisfy the constraint ρ≤1, by...

An integro-differential equation for 1D cell migration

Accounting for hard congestion in crowd motion modeling leads to non-smooth evolution problems. At the microscopic level (individuals are represented separately), these problems fit in the framework of non smooth analysis in Hilbert spaces, and the tools developed in the 70’s to handle the so-called sweeping process are directly adaptable. At the m...

Direct numerical simulations of the individual and collective dynamics of neutral squirmers are presented. “Squirmer” refers to a class of swimmers driven by prescribed tangential deformations at their surface, and “cneutral” means that the swimmer does not apply a force dipole on the fluid. The squirmer model is used in this article to describe se...

In this article, we propose an integrated model for oxygen transfer into the blood, coupled with a lumped mechanical model for the ventilation process. Objectives. We aim at investigating oxygen transfer into the blood at rest or exercise. The first task consists in describing nonlinear effects of the oxygen transfer under normal conditions. We als...

Use of cellular phones has been shown to be associated with crashes but many external distractions remain to be studied. To assess the risk associated with diversion of attention due to unexpected events or secondary tasks at the wheel. Responsibility case-control study. Adult emergency department of the Bordeaux University Hospital (France) from A...

In this note, we present a smooth extension method for the simulation of the motion of immersed rigid bodies. It is a method of the fictitious domain type, which uses Cartesian meshes and recovers the optimal order of the error by finding a smooth extension of the exact solution defined in the domain with holes. We first present the method with a P...

This book proposes an introduction to the mathematical modeling of the respiratory system. A detailed introduction on the physiological aspects makes it accessible to a large audience without any prior knowledge on the lung. Different levels of description are proposed, from the lumped models with a small number of parameters (Ordinary Differential...

This chapter addresses the question of oxygen transfer from air to blood. Section 5.1 gives a general overview of the phenomena on which transfer relies: diffusion through various barriers, and capturing of oxygen by hemoglobin. The notion of Diffusing Capacity is introduced. The next Section 5.2 presents a global model with a minimal set of variab...

This chapter gives a theoretical framework for the modeling of the respiratory tract as a resistive tree. We shall consider here the respiratory tree, or more general networks, as a collection of interconnected pipes, through which a viscous fluid is flowing according to Poiseuille’s law, so that the flux through an individual pipe depends linearly...

This chapter presents lumped models of the respiratory systems: a small number a variables of interest is selected, and the physical phenomena are modeled as differential equations on those variables. Section 2.1 details the simplest approach to model the ventilation as a linear ODE on the volume. It is based on two main parameters: resistance R an...

We investigate in this chapter the possibility to elaborate a mathematical model of the lung as an infinite resistive tree. The approach is an extension of Section 3. 3 (p. 74) to the case of infinite networks. In the infinite setting, the notions of boundary and functions on this boundary have to be designed with care. We give some general propert...

Inertia is known to play a significant role in the upper part of the respiratory tract. Additionally, an accurate description of the air velocity field in the branches can be useful in many situations, for example if one aims at investigating the deposition process of sprays. On the other hand, a full computation of the velocity field in all the re...

To assess the association between mind wandering (thinking unrelated to the task at hand) and the risk of being responsible for a motor vehicle crash. Responsibility case-control study. Adult emergency department of a university hospital in France, April 2010 to August 2011. 955 drivers injured in a motor vehicle crash. Responsibility for the crash...

- The Arbitrary Lagrangian-Eulerian framework allows to compute free surface flows with the Finite Element functions defined on a fittedmesh which follows the globalmotion of the fluid domain. We describe here how freefem++ can be used to implement this method, and we provide two and three dimensional illustrations in the context of water waves.

Background Despite progresses achieved in road safety the number of lives saved is reaching a plateau in developed countries. Inattention at the wheel remains understudied. Aims/Objectives/Purpose In order to examine new ideas about potential causal factors and targets for interventions we investigated the association between mind wandering (MW) an...

We are interested in the finite element solution of elliptic problems with a right-hand side of the single layer distribution type. Such problems arise when one aims at accounting for a physical hypersurface (or line, for bi-dimensional problem), but also in the context of fictitious domain methods, when one aims at accounting for the presence of a...

We propose a coupling strategy for solving efficiently bifluid flows based on the Stokes equations. Our approach relies on a level set formulation of the interface-capturing problem, and involves a finite element discretization for the fluid resolution, the method of characteristics for solving the advection of the interface and the anisotropic mes...

We present two-dimensional simulations of chemotactic self-propelled bacteria swimming in a viscous fluid. Self-propulsion is modelled by a couple of forces of same intensity and opposite direction applied on the rigid bacterial body and on an associated region in the fluid representing the flagellar bundle. The method for solving the fluid flow an...

The Fat Boundary Method is a method of the Fictitious Domain class, which was proposed to solve elliptic problems in complex geometries with non-conforming meshes. It has been designed to recover optimal convergence at any order, despite of the non-conformity of the mesh, and without any change in the discrete Laplace operator on the simple shape d...

This paper deals with a class of macroscopic models for cell migration in a saturated medium for two-species mixtures. Those species tend to achieve some motion according to a desired velocity, and congestion forces them to adapt their velocity. This adaptation is modelled by a correction velocity which is chosen minimal in a least-square sense. We...

We address here the issue of congestion in the modeling of crowd motion, in the non-smooth framework: contacts between people are not anticipated and avoided, they actually occur, and they are explicitly taken into account in the model. We limit our approach to very basic principles in terms of behavior, to focus on the particular problems raised b...

We propose here a general framework to address the question of trace operators on a dyadic tree. This work is motivated by the modeling of the human bronchial tree which, thanks to its regularity, can be extrapolated in a natural way to an infinite resistive tree. The space of pressure fields at bifurcation nodes of this infinite tree can be endowe...

We prescrit a method to simulate the motion of self-propelled rigid particles in a twodimensional Stokesian fluid, taking into account chemotactic behaviour. Self-propulsion is modelled as a point force associated to each particle, placed at a certain distance from its gravity centre. The method for solving the fluid flow and the motion of the bact...

A simple model to handle the flow of people in emergency evacuation situations is considered: at every point x, the velocity U(x) that individuals at x would like to realize is given. Yet, the incompressibility constraint prevents this velocity field to be realized and the actual velocity is the projection of the desired one onto the set of admissi...

We propose here a decomposition of the respiratory tree into three stages which correspond to different mechanical models. The resulting system is described by the Navier-Stokes equation coupled with an ODE (a simple spring model) representing the motion of the thoracic cage. We prove that this problem has at least one solution locally in time for...

The aim of this paper is to develop a model of the respiratory system. The real bronchial tree is embedded within the parenchyma, and ventilation is caused by negative pressures at the alveolar level. We aim to describe the series of pressures at alveolae in the form of a function, and to establish a sound mathematical framework for the instantaneo...

The aim of this paper is to develop a crowd motion model designed to handle highly packed situations. The model we propose rests on two principles: We first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other people; The actual velocity is then computed as the projection of the...

In this article we are interested in the simulation of the air *ow in the bronchial tree. The model we use has already been described by Baffico, Grandmont and Maury and is based on a three part description of the respiratory tract. This model leads, after time discretization, to a Stokes system with non standard dissipative boundary conditions tha...

We present here some contributions to the numerical analysis of the penalty method in the finite element context. We are especially interested in the ability provided by this approach to use Cartesian, non boundary-fitted meshes to solve elliptic problems in complicated domain. In the spirit of fictitious domains, the initial problem is replaced by...

We present a numerical tool which aims at investigating the rheology of dense suspensions of entities such as spheres, red blood cells, polymer chains, or any kind of rigid or deformable bodies, in a viscous fluid. We shall pay a special attention to the short-range interactions between those entities (contact forces, lubrication forces). As for th...

In a previous paper, we proposed a model for crowd motion, together with a numerical algorithm, especially designed to handle highly packed situations. This model rests oil two principles: We first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other peopled The actual velocity i...

Dans cette Note, nous présentons une méthode pour résoudre numériquement l'équation de Stokes modélisant l'écoulement incompressible de deux fluides non-miscibles ayant des viscosités très différentes. La résolution des systèmes d'équations provenant de la discrétisation par éléments finis est faite par la méthode d'Uzawa. Les problèmes de conditio...

We consider here a discrete system of spheres interacting through a lubrication force. This force is dissipative, and singular near contact: it behaves like the reciprocal of interparticle distance. We propose a macroscopic constitutive equation which is built as the natural continuous counterpart of this microscopic lubrication model. This model,...

We study the equations of motion of two immiscible fluids with comparable densities, but very different viscosities in a two-dimensional horizontal pipe. This is applied to the lubricated transportation of heavy crude oil. First, we write the problem in variational form and next we derive an energy balance for this model.

We propose here a decomposition of the respiratory tree into three stages which correspond to different mechanical models. The resulting system is described by the Navier-Stokes equation coupled with an ODE (a simple spring model) representing the motion of the diaphragm muscl. We prove that this problem has at least one solution locally in time fo...

We are interested in the impact of the mechanical characteristics of the bronchial tree and lung parenchyma on the gas exchange capacity of the human lungs. Objectives: The precise role of the airway smooth muscle (ASM) remains a matter of debate. ASM could have a structural function, to confer transport optimized mechanical properties to the bronc...

We present here a new framework to handle the short-range interaction between rigid bodies in a viscous, incompressible fluid. This framework is built as the vanishing viscosity limit of a lubrication model. We restrict ourselves here to the case of a single particle and a rigid wall. Our approach is based on a standard first-order approximation fo...

We discretize in space the equations obtained at each time step when discretizing in time a Navier-Stokes system modelling the two-dimensional flow in a horizontal pipe of two immiscible fluids with comparable densities, but very different viscosities. At each time step the system reduces to a generalized Stokes problem with nonstandard conditions...

We propose here a numerical scheme to compute the motion of rigid bodies with a non-elastic impact law. The method is based on a global computation of the reaction forces between bodies. Those forces, whose direction is known since we neglect friction effects, are identified at the discrete level with a scalar which plays the role of a Kuhn-Tucker...

In this paper we elaborate a model to describe some aspects of the human lung considered as a continuous, deformable, medium. To that purpose, we study the asymptotic behavior of a spring-mass system with dissipation. The key feature of our approach is the nature of this dissipation phenomena, which is related here to the flow of a viscous fluid th...

We investigate the behavior of fluid–particle mixtures subject to shear stress, by mean of direct simulation. This approach is meant to give some hints to explain the influence of interacting red cells on the global behavior of the blood. We concentrate on the apparent viscosity, which we define as a macroscopic quantity which characterizes the res...

We propose inflow and outflow boundary conditions to handle the airflow through the upper part of bronchial tree-like domains. This approach is based on a condensation of the part of the domain which is geometrically complex by using the linearity of the flow through a net of ducts with small diameters with respect to the pressure jumps. We show th...

The Fat Boundary Method (FBM) is a fictitious domain like method for solving partial differential equations in a domain with holes Ω ∖ [`(B)]\bar B — where B is a collection of smooth open subsets — that consists in splitting the initial problem into two parts to be coupled via Schwartz type iterations: the solution, with a fictitious domain approa...

We present here a method to simulate the motion of a rigid body in a fluid. The method is based on a variational formulation on the whole fluid/solid domain, with some constraints on the unknown and the test functions. These constraints are relaxed by introducing a penalty term, which leads to a minimization problem over unconstrained functional sp...

We present here an algorithm to simulate the motion of rigid bodies subject to a non-overlapping constraint, and which tend to aggregate when they get close to each other. The motion is induced by external forces. Two types of forces are considered here: drift force induced by the action of a surrounding fluid whose motion is prescribed, and stocha...

In Carlier et al. (ESAIM Proceedings, CEMRACS 1999), an algorithm was proposed to approximate the projection of a function f∈H10(Ω) (where Ω is a convex domain) onto the cone of convex functions. This algorithm is based on a dual expression of the constraint, which leads to a saddle-point problem which has no solution in general. We show here that...

Our purpose is to estimate numerically the influence of particles on the global viscosity of fluid–particle mixtures. Particles are supposed to rigid, and the surrounding fluid is newtonian. The motion of the mixture is computed directly, i.e. all the particle motions are computed explicitly. Apparent viscosity, based on the force exerted by the fl...

Motivated by the study of environmental fluids with massive particle sedimentation, we have used direct numerical simulations to study massive particles falling in a viscous fluid with periodic boundaries in a two-dimensional geometry. We computed different cases involving different particle concentration ranging from 0.02 to 0.4, and three differe...

We consider the Poisson equation with Dirichlet boundary conditions, in a domain Ω\ $\bar B$ , where Ω⊂ ℝn , and B is a collection of smooth open subsets (typically balls). The objective is to split the initial problem into two parts: a problem set in the whole domain Ω, for which fast solvers can be used, and local subproblems set in narrow doma...

To estimate parameters of concurrent sexual partnerships in five urban populations in sub-Saharan Africa and to assess their association with levels of HIV infection and other sexually transmitted infections (STI). Data were obtained from a multicentre study of factors which determine the differences in rate of spread of HIV in five African cities....

We describe an algorithm to approximate the minimizer of an elliptic functional in the form on the set of convex functions u in an appropriate functional space X. Such problems arise for instance in mathematical economics [4]. A special case gives the convex envelope of a given function . Let be any quasiuniform sequence of meshes whose diameter go...

We investigate numerical methods to approximate the projection-operator from H 0 1 into the set of convex functions. We introduce a new formulation of the problem, based on gradient fields. It leads in a natural way to an infinite-dimensional saddle-point problem, which can be shown to be ill-posed in general. Existence and uniqueness of a saddle p...

We describe an algorithm to approximate the minimizer of an elliptic functional in the form ∫Ωj(x,u,∇u) on the set C of convex functions u in an appropriate functional space X. Such problems arise for instance in mathematical economics [5]. A special case gives the convex envelope u0∗∗ of a given function u0. Let (Tn) be any quasi-uniform sequence...

We propose a method to simulate the motion of 2D rigid particles in a viscous, incompressible fluid. Within the arbitrary Lagrangian Eulerian framework, momentum equations for both the fluid and the particles are discretized, and a coupled variational formulation is established. By introducing an appropriate finite element approximation, a symmetri...

In [2], we presented direct numerical simulations of the motion of solid particles in a viscous fluid. In such simulations, numerical problems are likely to occur when particles get very close to each other: the mesh is to be refined in the gap zone, which is computationally expensive. In order to overcome this problem, we introduce here a lubricat...